KFUPM-1

1
Iterative Computer Algorithmsand their
applications in engineering

• Sadiq M. Sait, Ph.D
• sadiq_at_kfupm.edu.sa
• Department of Computer Engineering
• King Fahd University of Petroleum and Minerals
• Dhahran, Saudi Arabia
• Special Talk for Motorola CAD Group, Austin

2
Talk outline

• The various general iterative non-deterministic
  algorithms for combinatorial optimization.
• Search, examples of hard problems
• SA, TS, GA, SimE and StocE
• Their background and operation
• Parameters
• Differences
• Applications
• Some research problems and related issues
  Convergence, parallelization, hybridization,
  fuzzification, etc.

3
Talk outline

• Based on the text book by Sadiq M. Sait and
  Habib Youssef entitled Iterative Computer
  Algorithms and their applications in engineering
  to be published by IEEE Computer Society Press,
  1999.

4
Terminology

• Combinatorics Does a particular arrangement
  exist?
• Combinatorial optimization Concerned with the
  determination of an optimal arrangement or order
• Hard problems NP NP complete.
• Examples QAP, Task scheduling, shortest path,
  TSP, partitioning (graphs, sets, etc), HCP, VCP,
  Topology Design, Facility location, etc
• Optimization methods
• Constructive Iterative
• Aim at improving a certain cost function

5
Examples

• QAP Required to assign M modules to L locations
  (LgtM), in order to minimize a certain objective
• wire-length, timing, dissipation, area
• Number of solutions is given by L!
• Task Scheduling Given a set of tasks (n)
  represented by an acyclic DAG, and a set of
  inter-connected processors (m), it is required to
  assign the tasks to processors in order to
  minimize the time to completion of the tasks.
• Number of solutions given by mn

6
QAP
7
Scheduling
8
Purpose

• To motivate application of iterative search
  heuristics to hard practical engineering
  problems.
• To understand some of the underlying principles,
  parameters, and operators, of these modern
  heuristics.

9
Terminology

• Search space
• Move (perturb function)
• Neighborhood
• Non-deterministic algorithms
• Optimal/Minimal solution

10
Simulated Annealing

• Most popular and well developed technique
• Inspired by the cooling of metals
• Based on the Metropolis experiment
• Accepts bad moves with a probability that is a
  decreasing function of temperature
• Prob(accept) exp(?E)/KT
• E represents energy (cost)

11
The Basic Algorithm

• Start with
• a random solution
• a reasonably high value of T (problem dependent)
• Call the Metropolis function
• Update parameters
• Decrease temperature (T?)
• Increase number of iterations in loop, i.e., M,
  (M?)
• Keep doing so until freezing, or, out of time

12
Metropolis Loop

• Begin Loop Generate a neighbor solution
• Compute difference in cost between old and
  neighboring solution
• If costlt0 then accept, else accept only if
• random lte?cost/T
• Decrement M, repeat loop until M0

13
Parameters

• Also known as the cooling schedule
• Comprises
• choice of proper values of initial temperature To
  
• decrement factor ?lt1
• parameter ?gt1
• M (how many times the Metropolis loop is
  executed)
• stopping criterion

14

Characteristics

• Given enough time it will converge to an optimal
  state
• Very time consuming
• During initial iterations, behaves like a random
  walk algorithm, during later iterations it
  behaves like a greedy algorithm, a weakness
• Very easy to implement
• Parallel implementations available

15

Requirements

• Requirements
• A representation of the state
• A cost function
• A neighbor function
• A cooling schedule
• Time consuming steps
• Computation of cost due to move must be done
  efficiently (estimates of costs are used)
• Neighbor function may also be time consuming

16
Applications

• Has been successfully applied to a large number
  of combinatorial optimization problems in
• science
• engineering
• medicine
• business
• etc

17
Genetic Algorithms

• Introduced by John Holland and his colleagues
• Inspired by Darwinian theory of evolution
• Emulates the natural process of evolution
• Based on theory of natural selection
• that assumes that individuals with certain
  characteristics are better able to survive
• Operate on a set of solutions (termed population)
• Each individual of the population is an encoded
  string (termed chromosome)

18
Genetic Algorithms

• Strings (chromosomes) represent points in the
  search space
• Each iteration is referred to as generation
• New sets of strings called offsprings are created
  in each generation by mating
• Cost function is translated to a fitness function
• From the pool of parents and offsprings,
  candidates for the next generation are selected
  based on their fitness

19
Requirements

• To represent solutions as strings of symbols or
  chromosomes
• Operators To operate on parent chromosomes to
  generate offsprings (crossover, mutation,
  inversion)
• Mechanism for choice of parents for mating
• A selection mechanism
• A mechanism to efficiently compute the fitness

20
Operators

• Crossover The main genetic operator
• Types Simple, Permutation based (such as Order,
  PMX, Cyclic), etc.
• Mutation To introduce random changes
• Inversion Not so much used in applications

21
Crossover

• Example
• Chromosome for the scheduling problem of
  eight tasks, to be assigned to three processors
  
• 1 2 3 1 3 1 1 2 , 1 2 3 3 1 3 2 2 (index
  of the array refers to the task, and the value
  the processor it is assigned to)

22
Simple Crossover

• Cut and catenate
• Let the crossover point be after task 5, as
  shown. Then the offspring created by the simple
  crossover will be as follows
• Chromosome for the scheduling problem of 8 tasks
  to be assigned to three processors
• Parent 1 1 2 3 1 3 ? 1 1 2
• Parent 2 1 2 3 3 1 ? 3 2 2
• Offspring generated 1 2 3 1 3 3 2 2

23
Permutation Crossovers

• Consider the linear placement problem of 8
  modules (a, b, ...,g, h,) to 8 slots.
• Parent 1 h d a e b ? c g f
• Parent 2 d b c g a ? f h e
• Offspring generated h d a e b f h e
• The above offspring is not a valid solution
  since modules e and h are assigned to more than
  one location, and modules c, and g are lost

24
Order Crossovers

• Parent 1 h d a e b ? c g f
• Parent 2 d b f c a ? g h e
• Offspring generated h d a e b ? f c g
• The above offspring represents a valid solution

25
Mutation

• Similar to the perturb function used in simulated
  annealing.
• The idea is to produce incremental random changes
  in the offsprings
• Important, because crossover is only an
  inheritance mechanism, and offsprings cannot
  inherit characteristics which are not in any
  member of the population.
• Size of the population is generally small.

26
Mutation

• Example Consider the population below
• s1 0 1 1 0 0 1
• s2 1 0 1 1 0 0
• s3 1 1 0 1 0 1
• s4 1 1 1 0 0 0
• Observe that the second last gene in all
  chromosomes is always 0, and the offsprings
  generated by simple crossover will never get a 1.

27
Decisions to be made

• What is an efficient chromosomal representation?
• Probability of crossover (Pc)? Generally close
  to 1
• Probability of mutation (Pm) kept very very
  small, 1 - 5 (Schema theorem)
• Type of crossover? and, what mutation scheme?
• Size of the population? How to construct the
  initial population?
• What selection mechanism to use, and the
  generation gap (i.e., what percentage of
  population to be replaced during each generation?)

28
Problems

• Mapping cost function to fitness
• Premature convergence can occur. Scaling methods
  are proposed to avoid this
• Requires more memory and time
• Several parameters, and can be very hard to tune

29
Applications

• Classical hard problems (TSP, QAP, Knapsack,
  clustering, N-Queens problem, the Steiner tree
  problem, Topology Design, etc.,)
• Problems in high-level synthesis and VLSI
  physical design,
• Others such as
• Scheduling,
• Power systems, telecommunications (maximal
  distance codes, telecom NW design), etc.
• Fuzzy control (GAs used to identify fuzzy rule
  set)

30
Example

• aghcbidef is a possible chromosome

31
Some Variations

• 2-D chromosomes
• Gray versus Binary encoding
• Multi-objective optimization with GAs
• Constant versus dynamically decreasing population
• Niches, crowding and speciation
• Scaling
• etc

32
Tabu Search

• Introduced by Fred Glover
• Generalization of Local Search
• At each step, the local neighborhood of the
  current solution is explored and the best
  solution is selected as the next solution
• This best neighbor solution is accepted even if
  it is worse than the current solution (hill
  climbing)

33
Central Idea

• Exploitation of memory structures
• Short term memory
• Tabu list
• Aspiration criterion
• Intermediate memory for intensification
• Long term memory for diversification

34
Basic Short-Term TS

• 1. Start with an initial feasible solution
• 2. Initialize Tabu list and aspiration level
• 3. Generate a subset of neighborhood and find the
  best solution from the generated ones
• 4. If move in not in tabu list then accept
• else
• If move satisfies aspiration criterion then
  accept
• 5. Repeat above 2 steps until terminating
  condition

35
Intensification/Diversification

• Intensification Intermediate term memory is used
  to target a specific region in the space and
  search around it thoroughly
• Diversification Long term memory is used to
  store information such as frequency of a
  particular move, etc., to take search into
  unvisited regions.

36
Implementation related issues

• Size of candidate list?
• Size of tabu list?
• What aspiration criterion to use?
• Fixed or dynamic tabu list?
• What intensification strategy?
• What diversification scheme to use?
• And several others

37
Tabu list and Move Attributes

• Moves or attributes of moves are stored in tabu
  lists (storing entire solutions is expensive)
• Tabu list size is generally small (short-term)
• Tabu list size may be fixed or changed
  dynamically
• Possible data structures are queues and arrays

38
Related Issues

• Design of evaluator functions
• Candidate list strategies
• Target analysis
• Strategic oscillation
• Path relinking
• Parallel implementation
• Convergence aspects
• Applications (again several)

39
Simulated Evolution

• Like GAs, also mimics biological evolution
• Each element of the solution is thought of as an
  individual with some fitness (goodness)
• The basic procedure consists of
• evaluation
• selection, and,
• allocation
• Based on compound moves

40
Evaluation

• Goodness is defined as the ratio of optimal cost
  to the actual cost
• ?i , gioi /ci
• Selection is based on the goodness of the element
  of a solution
• The optimal cost is determined only once
• The actual cost of some individuals changes with
  each iteration

41
Selection

• Selection The higher the goodness value, higher
  the chance of the module staying in its current
  location
• Pi min(1gi ,1)
• where gi is the goodness of element i.
  That is, low goodness maps to a high probability
  of the module being altered.
• The selection operator has a non-deterministic
  nature and this gives SimE the hill climbing
  capability
• Selection is generally followed by sorting

42
Allocation

• This is a complex form of genetic mutation
  (compound move)
• This operator takes two sets (selection S and
  remaining set R) and generates a new population
• Has the most impact on the rate of convergence

43
Comparison of SimE and SA

• In SA a perturbation is a single move
• For SA, the elements to be moved are selected at
  random
• SA is guided by a parameter called temperature,
  while for SimE the search is guided by the
  individual fitness of the solution components

44
Comparison of SimE and GA

• SimE works on a single solution called population
  while in GA, the set of solutions comprises the
  population
• GA relies on genetic reproduction (using
  crossover, mutation, etc).
• In SimE, an individual is evaluated by estimating
  the fitness of each of its genes. (Genes with
  lower fitness have a higher probability of
  getting altered)

45
Other facts

• Fairly simple, yet very powerful
• Has been applied to several hard problems (such
  as VLSI standard cell placement, high level
  synthesis, etc)
• Parallel implementations have been proposed (for
  MISD and MIMD)
• Convergence analysis presented by designers of
  the heuristic and others

46
Stochastic Evolution

• StocE, often confused with Simulated Evolution
• Distinguishing features
• The probability of accepting a bad move increases
  if no good solutions are found
• Like SimE, is based on compound moves (perturb
  function)
• There is a built in mechanism to reward the
  algorithm whenever a good solution is found

47
Parameters Inputs

• An initial solution So
• An initial value of control parameter po
• Gain (m) gt RANDINT(-p,0) (accepting both good
  and poor solutions)
• Stopping criterion parameter called R

48
Functions

• PERTURB To make a compound move to a new state.
• UPDATE function p p incr (p) is incremented
  to allow uphill moves
• Infeasible solutions are accepted, and then a
  function MAKESTATE is invoked to undo some last k
  moves.

49
Comparison of StocE and SA

• In StocE a perturbation is a compound move
• There is no hot and cold regime
• In SA, the acceptance probability keeps
  decreasing with time (decreasing values of
  temperature)
• StocE introduces the concept of reward whereby
  the search algorithm cleverly rewards itself
  whenever a good move is made

50
Common features of All heuristics

• All are general iterative heuristics, can be
  applied to any combinatorial optimization problem
• All are conceptually simple and elegant
• All are based on moves and neighborhood
• All are blind
• All occasionally accept inferior solutions (i.e,
  have hill-climbing capability)
• All are non-deterministic (except TS which is
  only to some extent)
• All (under certain conditions) asymptotically
  converge to an optimal solution (TS and StocE)

51
Some Research Areas

• Applications to various hard problems of current
  technology?
• Hybridization?
• How to enhance strengths and compensate for
  weaknesses of two or more heuristics
• Examples SA/TS, GA/SA, TS/SimE, etc
• Fuzzy logic for multi-objective optimization and
  other operations.
• Parallel implementations
• Convergence aspects